一类波动方程解的生命跨度的估计

Upper Bound Lifespan Estimates of Solution to a Wave Equation

在外区域上研究一类变系数且带组合非线性项的波动方程,其中空间维数满足n(n≥2)。首先,分别考虑带幂次非线性项|u|^(p)与导数非线性项|u_(t)|^(p)的问题。运用检验函数方法和Kato引理,导出Riccati方程,证明了解会破裂,结合叠加原理给出其生命跨度的上界估计。其次,选取适当的检验函数,采用Kato引理即给出生命跨度的上界估计。通过比较,给出生命跨度的最佳上界估计。

This paper is concerned with the wave equation with variable coefficient and combined nonlinearities on exterior domain,where the space dimension satisfies n(n≥2).Firstly,the problems with power nonlinearity|u|^(p)and derivative nonlinearity|u_(t)|^(p)are considered,respectively.The blow-up of solutions to the problems are established by applying test function method,the Kato lemma and the Riccati equation.Then,the upper bound lifespan estimate of solution is derived by using superposition principle.Secondly,upper bound lifespan estimate of solution is investigated by making use of suitable test function and the Kato lemma.The optimal upper bound lifespan estimate is obtained by comparing with the two results.